Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set - Page 599: 66


$x\lt-1 \text{ or } x\gt9$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Solve the given inequality, $ |x-4|+5\gt10 ,$ by isolating first the absolute value expression. Then use the definition of a greater than absolute value inequality. Finally, graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the properties of inequality, the given expression is equivalent to \begin{array}{l}\require{cancel} |x-4|+5\gt10 \\\\ |x-4|\gt10-5 \\\\ |x-4|\gt5 .\end{array} Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the given inequality is equivalent to \begin{array}{l}\require{cancel} x-4\gt5 \\\\\text{OR}\\\\ x-4\lt-5 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} x-4\gt5 \\\\ x\gt5+4 \\\\ x\gt9 \\\\\text{OR}\\\\ x-4\lt-5 \\\\ x\lt-5+4 \\\\ x\lt-1 .\end{array} Hence, the solution set is $ x\lt-1 \text{ or } x\gt9 .$
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